Currently working for Company A who applied for my h-1b in Nov and Got approved. Now I have permanent job offer from Company B who would like to apply for H-1b transfer. Company B would like to apply my H-1b in Jan but want me to start working from Feb

Q1. Is it safe to change the employer right after getting approval from Company A.
Q2. If I get H-1b approval(Premium Processing) from Company B in Jan. Can I still work for Company A until Jan end and join Company B in Feb.
Q3. Can Company A put me in trouble since they paid for my h-1b. I didn't sign any contract.